Question

Answer the following questions based on the accompanying diagram.

1. How much would the firm’s revenue change if it lowered the price from$\mathbf{\backslash (}\mathbf{12}\mathbf{}\mathbf{to}\mathbf{}\mathbf{\backslash )}\mathbf{10}$? Is demand elastic or inelastic in this range?
2. How much would the firm’s revenue change if it lowered the price from$\mathbf{\backslash (}\mathbf{4}\mathbf{}\mathbf{to}\mathbf{}\mathbf{\backslash )}\mathbf{2}$? Is demand elastic or inelastic in this range?
3. What price maximizes the firm’s total revenues? What is the elasticity of demand at this point on the demand curve?

Short answer

1. The revenue would increase by $8. The value of elasticity of demand is $-3.665$which is greater than 1. Hence the demand is elastic.
2. The revenue would decrease by $8. The value for elasticity of demand value is$-0.27$ which is less than 1. Hence, demand is inelastic
3. At price of$\mathrm{\$}7$ the firm maximizes its revenue and at that point demand is unitary elastic.

Step-by-step solution

To calculate firm’s revenue when price is lowered from $12 to $10

a.

To get the firm’s revenue by the equation as$\mathrm{TR}={\mathrm{P}}_{\mathrm{x}}{\mathrm{Q}}_{\mathrm{x}}$

If the price is $ $12$, and the output quantity is$1$

$\begin{array}{c}\mathrm{TR}={\mathrm{P}}_{\mathrm{x}}{\mathrm{Q}}_{\mathrm{x}}\\ =12\times 1\\ \mathrm{TR}=12\end{array}$

Thus, the total revenue of the price is $$12$.

If the price is $$10$, the output quantity is 2 units.

$\begin{array}{c}\mathrm{TR}={\mathrm{P}}_{\mathrm{x}}{\mathrm{Q}}_{\mathrm{x}}\\ =10\times 2\\ \mathrm{TR}=20\end{array}$

Thus, the total revenue of the price is $20.

The revenue would increase by$(\mathrm{\$}20-\mathrm{\$}12)=\mathrm{\$}8$.

If the demand is elastic or inelastic by the equation as

${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{x}}}=\left(\frac{{\mathrm{Q}}_2-{\mathrm{Q}}_1}{{\mathrm{P}}_2-{\mathrm{P}}_1}\right)\times \left(\frac{\frac{{\mathrm{P}}_1+{\mathrm{P}}_2}{2}}{\frac{{\mathrm{Q}}_1+{\mathrm{Q}}_2}{2}}\right)$

Substitute the given data values are ${\mathrm{Q}}_2=2,\mathrm{\$}10,{\mathrm{Q}}_1=1.{\mathrm{P}}_2=\mathrm{\$}10,{\mathrm{P}}_1=\mathrm{\$}12$

$\begin{array}{c}{\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{x}}}=\left(\frac{2-1}{10-12}\right)\times \left(\frac{\frac{12+10}{2}}{\frac{1+2}{2}}\right)\\ =\left(\frac{1}{-2}\right)\times \left(\frac{\frac{22}{2}}{\frac{3}{2}}\right)\\ =-0.5\times \frac{11}{1.5}\\ =-0.5\times 7.33\\ {\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{x}}}=(-3.665)\end{array}$

The value for elasticity of demand value is $(-3.665)$ which is greater than 1 thus, demand is elastic.

To calculate firm’s revenue lowered the price from $4 to $2:

b.

To get the firm’s revenue by the equation as

If the price is $ $4$, the output quantity is 5 units.

$\begin{array}{c}\mathrm{TR}={\mathrm{P}}_{\mathrm{x}}{\mathrm{Q}}_{\mathrm{x}}\\ =4\times 5\\ \mathrm{TR}=20\end{array}$

Total revenue is $ $20$.

If the price is $$2$, the output is 6 units.

$\begin{array}{c}\mathrm{TR}={\mathrm{P}}_{\mathrm{x}}{\mathrm{Q}}_{\mathrm{x}}\\ =2\times 6\\ \mathrm{TR}=12\end{array}$

Hence, the total revenue is $$12$.

The revenue will decrease by$(\mathrm{\$}12-\mathrm{\$}20)=-\mathrm{\$}8$.

If the demand is elastic or inelastic by the equation as

${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{x}}}=\left(\frac{{\mathrm{Q}}_2-{\mathrm{Q}}_1}{{\mathrm{P}}_2-{\mathrm{P}}_1}\right)\times \left(\frac{\frac{{\mathrm{P}}_1+{\mathrm{P}}_2}{2}}{\frac{{\mathrm{Q}}_1+{\mathrm{Q}}_2}{2}}\right)$

Substitute the given data values are${\mathrm{Q}}_2=6,\mathrm{\$}2,{\mathrm{Q}}_1=5,{\mathrm{P}}_2=\mathrm{\$}2,{\mathrm{P}}_1=\mathrm{\$}4$

$\begin{array}{c}{\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{x}}}=\left(\frac{6-5}{2-4}\right)\times \left(\frac{\frac{4+2}{2}}{\frac{5+6}{2}}\right)\\ =\left(\frac{1}{-2}\right)\times \left(\frac{\frac{2}{11}}{\frac{1}{2}}\right)\\ =-0.5\times \frac{3}{5.5}\\ =(-27)\end{array}$

The value for elasticity of demand value is ( $-0.27$ ). Since, the value of elasticity of demand is less than 1, the demand is inelastic.

To determine price maximizing revenue:

c.

To determine the price for maximization of the firm’s total revenue, look at the graph. The marginal revenue between the marginal axis and the demand curve is the area. The price starts at$\mathrm{\$}14$ and intersects the quantity is$3.5$ the marginal revenue is$0$ when the quantity is$3.5$ and the price is $\$7$.If the elasticity of demand maximum is equal to$1$ and revenue is $0$.

If the price is $7, it maximizes the total firm’s revenues and at that point of elasticity, demand is unitary elastic.